Sectoral Financial Balance Definitions & Data

Private Financial Balance (aka Private Net Savings):  Here is a  good definition  from economist Scott Fullwiler.  "Private sector financial balance or Private sector net savings is the addition/subtraction to net financial wealth for the private sector in a given period. If the private sector is net borrowing, then its balance will be negative (deficit); if it is net saving, then its balance will be positive (surplus)".^[1][2]

Government Financial Balance: Obviously, when negative we have a budget deficit, if positive we have a budget surplus.^[3]

Current Account Balance: The current account is the sum of the balance of trade (exports - imports of goods and services), net factor income (such as interest and dividends) and net transfer payments (such as foreign aid). In the US its largest component is the trade balance (exports - imports).  For example, if we import $50 of goods and export $30 of goods our current account balance is in deficit by $20 (assuming other current account components have zero balances).^[4][5] See here for more.

Capital Account Balance (aka Financial Account Balance): The capital account is the change in the foreign ownership of domestic assets minus the change in domestic ownership of foreign assets. In the US it is indeed mostly the financial account (which deals with only financial asset net changes). It is essentially the negative of the current account balance. In our example, the foreign sector as a whole now has a $20 surplus, with which they can buy dollar denominated assets such as US Treasuries or US buildings. So there will be a net change in foreign ownership of US assets by $20. In reality though the current account and the capital account balances don't exactly equal due to differences in accounting conventions and exchange rate fluctuations.^[4] See here for more.

Examples

Now let us take a look at some examples using the sectoral balances equation.  Real world data fits pretty close to the predicted equation in every case.

Private Financial _Balance = Government _Deficit + Current Account _Balance

If we look at data for 2009. The private balance is in surplus or the private sector is net saving, as the large government deficit more than offsets the current account deficit. This would be an addition to private sector net financial wealth by $1244 billion or 8.9% of GDP.

1243.7B = 1621.7B + -377.4B  (as % of GDP:  8.9 = 11.6 - 2.7 )

If we look at data for 2006.  The private balance is in deficit or the private sector is net borrowing, as the government deficit was not big enough to offset the large current account deficit. This would be a subtraction to private sector net financial wealth by $509 billion, 3.8% of GDP.

-508.9B = 291.6 + -798.4B  (as % of GDP:  -3.8 = 2.1 - 5.9)

If we look at data for 2000.  The private balance is once again in deficit as the government is running a surplus and the current account is in deficit.  Once again we have a subtraction to private sector net financial wealth by $557 billion, 5.6% of GDP.

-557.3B = -146.6B + -410.4B  (as % of GDP: -5.6 = -1.5 - 4.1) 

This data shows why the federal government should never run a budget surplus unless it is offset by a current account surplus.

Notes

1. See Line 37 here for private balance. Note that the definition here is NOT the same as in Line 3, see notes 2 .  
2. Line 3 is disposable income minus final consumption expenditures, it does not include investment expenditures on physical capital. For example, household sector (a part of the private sector)  final consumption expenditures is the 'C' component of GDP.  New housing purchases are not included in 'C', they are counted under investment 'I'. So Line 3 minus net physical capital formation (such as new houses) due to investment spending will give us the private financial balance.
3. See Line 40 here for government balance
4. Current, capital, financial accounts have different meanings in different contexts in national accounting. Here we are talking about foreign transaction current account, foreign transactions capital account etc:
5. See line 29 here for US current account balance

Why US GDP is all income made in the US

A country's GDP is all gross domestic income made in the country.  Those with some economics training may already know why this is so, but I would presume that most people are not aware of this.  If you look at US gross domestic income (GDI):  individual wages & salaries + corporate profits + proprietor income + rental income + net interest + indirect taxes on production + consumption of fixed capital ^[1][2], that figure comes out to be the same as the GDP.   Not exactly the same, no measurement is perfect, they are usually within 1%.  In 2009 US GDI was $13.862 trillion, while GDP was $13.939 trillion.

What is GDP?

GDP or Gross Domestic Product is the market value of all final goods and services produced within the borders of a country in a given period. There are 3 approaches  to calculate GDP, the most well known being the expenditure method,  i.e: measure all expenditures (spending) in the economy:  For spending to occur on a product, obviously it has to be produced.  Thus by measuring total spending we can measure total output or production.  See wiki for additional info.

GDP = C + G + I + (X - M)

C:  Personal  consumption expenditures: Total consumer spending on final goods and services. Makes up the lion share of US GDP at 71% in 2009
G:  Government consumption expenditures and gross investment: Total government spending on final goods and services.  Transfer payments such as social security and unemployment are not counted in G. A social security check is spend by the consumer who receives it, so it falls under C. Makes up about 20% of US GDP.
I:  Gross private domestic investment:  Mostly business investment in physical capital  (eg: software, factories, equipment). Financial investment is never counted, as that is saving ^{[3] [4]}. Also includes new residential housing purchases, and changes in private business inventories.  Makes up about 12% of US GDP in 2009, typically though it is closer to 16% during normal economic times.
X:  Exports are good/services produced domestically but not consumed domestically.  They must be added to get a true account of domestic production.
M:  Imports are goods/services not produced domestically but consumed domestically. They are already included in C, G or I, so it must be subtracted from GDP.  Since the US runs a trade deficit (X-M) is negative and thus takes away from GDP at -3%.

GDP as gross domestic income

GDP is also total gross domestic income, as one's spending is someone else's income. If you buy an IPod from Apple, that money isn't lit on fire. It is income for Apple's employees, management, investors, suppliers etc:.  Not only are you a consumer, you are a producer.  Your income comes from your role in the production of a good or service.  In other words the spending of another entity on that particular product results in your income. And this is true for all income earners. Thus all spending in the economy is also all income generated in the economy.

As a simple illustration, let us look at an economy with just 2 people. Let us say Jim hunts fish, and Alice farms potatoes.  During the year Jim "produces" 100 fish and sells them to Alice for $5 each. Alice "produces" 200 potatoes and sells them to Jim for $2 each.  The total GDP for the year in this economy is 100 X 5 + 200 X 2 = $900.  The total gross income made by Jim is $500, by Alice $400, for a total of $900. Of course the gross income minus spending is much smaller or negative in the case of Alice. But the point is the Alice's spending is Jim's gross income and vice versa.  The same is true in real world economies.  The spending of consumers, business and government is income for other business and consumers.  See pages 2-5 here (pdf) for an excellent explanation, from the US Dept. of Commerce.

Notes:

1. Consumption of fixed capital (economic depreciation) is included, as both GDI and GDP are gross figures.  Depreciation is essentially accounting for the portion of physical capital that wears out during a given period.  For Net Domestic Income (NDI) and Net Domestic Product (NDP) depreciation is subtracted.
2. Note that transfer payments such as social security income and welfare payments are not included in GDI. Given that they are transfers, this would result in double counting. Only incomes generated from the production of goods and services are included.
3. In economics saving is defined as not spending (deferred consumption).  By default any income made is saved, that is until it is spent.  Putting your money under a mattress, in a bank account or financial investment is all considered saving, as all these activities takes money out of the real economy.  Now if  money leaves financial institutions through other channels it may result in spending in the real economy, unless it is saved again. 
4. Let us take stocks as an example to illustrate the saving concept.  Buying stock is a transfer of claims on future production, not actual production. Which is why financial investment is never counted in GDP, as GDP is measure of production. Vast majority of stock purchases are third party transfers. If you buy IBM stock, you purchase it from a third party who already owns the stock, not IBM. Money never leaves the financial system. Only on an new share issue does IBM get any financial capital.  But even this is not counted in GDP, as IBM may choose to retain this money or buy  financial products, both of which are saving.  Only when that money is actually spend in the real economy does it count towards GDP.